Solutions to Ricci flow whose scalar curvature is bounded in L^p (II)

如果标量曲率在 L^p 中保持全局或局部限制（对于 p>= n/2），我们计划研究 n 维 Ricci 流的有限时间奇点结构。在该项目第一个资助期产生的预印本中，申请人和博士后刘嘉伟博士概括了申请人早期工作的成果。我们在此预印本中表明，如果解定义在封闭的 Kähler 2n 维（复 n 维）流形或实四维封闭流形上，则：a) 空间中的里奇曲率的积分 L^q 范数和时空中的 L^s 范数有界，其中 q 和 s 是取决于 n 和 p 的显式常数。 b) 黎曼曲率的 L^2 范数是有界的。我们进一步证明 a) 和 b) 的估计可以在 Kähler 设置中局部化。 在四个维度上，我们计划使用申请人早期工作的这些积分估计和浓度紧致性方法，并在适当的情况下使用 Bamler 和 Bamler-Zhang 的方法，以证明存在局部（或实际情况中的全局）双倍极限，因为 t-->T，T 是奇异时间。在 Kähler 局部环境中，我们计划在一般维度上进一步开发这些方法，并证明存在局部度量空间极限，因为 t-->T，T 是奇异时间，并且研究其结构。我们计划在四个维度上开发局部环折里奇流和在更高维度上开发局部圆锥形里奇流或其他局部弱里奇流，以便在奇异时间之后继续流动。

We plan to investigate the structure of finite time singularities of the Ricci flow in n dimensions if the scalar curvature remains globally or locally bounded in L^p for p>= n/2. In a preprint resulting from the first funding period of the project, the applicant and the Post-Doc Dr. Jiawei Liu, generalised results of earlier work of the applicant. We show in this preprint that if the solution is defined on a closed Kähler 2n-dimensional ( complex n-dimensional ) manifold, or a real four dimensional closed manifold, then:a) the integral L^q norm of of the Ricci curvature in space respectively L^s norm in space-time is bounded where q and s are explicit constants depending on n and p. b) the L^2 norm of the Riemannian curvature is bounded.We further show that the estimates of a) and b) can be localised in the Kähler setting. In four dimensions, we plan to use these integral estimates and concentration compactness methods of earlier work of the applicant, and where appropriate those of Bamler and Bamler-Zhang, to show that there is a local (or global in the real case) orbifold limit as t --> T, T being the singular time.In the Kähler local setting we plan to develop further these methods in general dimensions, and prove that there is a local metric space limit as t-->T, T being the singular time, and to study the structure thereof.We plan to develop a local orbifold Ricci flow in four dimensions and a local conical Ricci flow or other local weak Ricci flow in higher dimensions in order to continue the flow past the singular time.